Basic Electrical Fundamentals and Circuit Analysis

Fundamentals of Electrical Circuits

• Electric Circuit Definition: An electric circuit is an interconnection of electrical elements designed to communicate or transfer energy from one point to another.

• Fundamental Quantities:

  • Charge ($q$): An electrical property of atomic particles, measured in coulombs ($\text{C}$).
  • Current ($i$ or $I$): The time rate of change of charge, measured in amperes ($\text{A}$).
  • Mathematical definition:       $1\,\text{ampere} = 1\,\text{coulomb/second}$
  • Direct Current (DC): Current that remains constant with respect to time.
  • Alternating Current (AC): Current that varies sinusoidally with respect to time.
  • Voltage ($v$ or $V$): Also known as potential difference, it is the energy required to move a unit charge through an element, measured in volts ($\text{V}$).
  • Mathematical definition:       $1\,\text{volt} = 1\,\text{joule/coulomb} = 1\,\text{newton-meter/coulomb}$
  • Power ($p$): The time rate of expending or absorbing energy, measured in watts ($\text{W}$).
  • Energy ($w$): The capacity to do work, measured in joules ($\text{J}$).
  • Electric power utilities measure energy in watt-hours ($\text{Wh}$), where:       $1\,\text{Wh} = 3,600\,\text{J}$

• Law of Conservation of Energy: The algebraic sum of power in an electric circuit at any instant of time must be zero:   $\sum p = 0$

• Circuit Elements:

  • Ideal Independent Source: An active element that provides a specified voltage or current completely independent of all other circuit elements.     
  • Ideal Dependent (Controlled) Source: An active element in which the source quantity is controlled by another voltage or current within the circuit.     
  • Four possible configurations:
    1. Voltage-controlled voltage source (VCVS)
    2. Current-controlled voltage source (CCVS)
    3. Voltage-controlled current source (VCCS)
    4. Current-controlled current source (CCCS)

Ohm's Law and Circuit Operating States

• Ohm's Law: States that the voltage ($v$) across a resistor is directly proportional to the current ($i$) flowing through it:   $v \propto i$

  • Introducing resistance ($R$) as the constant of proportionality:     $v = i \times R$

• Resistance ($R$): Denotes an element's physical ability to resist the flow of electric current; measured in ohms ($\Omega$). Resistance is an inherent material property affected by internal and external conditions such as temperature.

• Conductance ($G$): The reciprocal of resistance, measuring an element's ability to conduct electric current; measured in siemens ($\text{S}$) or mho ($\mho$):   $G = \frac{1}{R}$

• Circuit States:

  • Short Circuit (SC): A circuit element with resistance approaching zero ($R = 0$).     
  • Conditions in SC: Voltage $v = 0$, Current $i = \infty$
  • Open Circuit (OC): A circuit element with resistance approaching infinity ($R = \infty$).     
  • Conditions in OC: Current $i = 0$, Voltage $v = \infty$ (or non-zero potential difference)

• Worked Example: Application of Ohm's Law

  • Problem: In a circuit where a $30\,\text{V}$ source is connected directly across a $5\,\text{k}\Omega$ resistor, calculate current ($i$), conductance ($G$), and power ($p$).
  • Current Calculation:     $i = \frac{v}{R} = \frac{30}{5 \times 10^3} = 6 \times 10^{-3}\,\text{A} = 6\,\text{mA}$
  • Conductance Calculation:     $G = \frac{1}{R} = \frac{1}{5 \times 10^3} = 0.2 \times 10^{-3}\,\text{S} = 0.2\,\text{mS}$
  • Power Calculation (verified via three methods):
  • Using voltage and current:       $p = v \times i = 30 \times (6 \times 10^{-3}) = 180\,\text{mW}$
  • Using current and resistance:       $p = i^2 \times R = (6 \times 10^{-3})^2 \times (5 \times 10^3) = 180\,\text{mW}$
  • Using voltage and conductance:       $p = v^2 \times G = (30)^2 \times (0.2 \times 10^{-3}) = 180\,\text{mW}$

Network Topology

• Topology Definitions:

  • Branch ($b$): Represents a single two-terminal element such as a voltage source, current source, or resistor.
  • Node ($n$): The point of connection between two or more branches.
  • Loop ($l$): Any closed path in a circuit.

• Fundamental Theorem of Network Topology: For any network containing $b$ branches, $n$ nodes, and $l$ independent loops:   $b = l + n - 1$

• Element Connections:

  • Series: Two or more elements are in series if they exclusively share a single node, resulting in the exact same current flowing through them.
  • Parallel: Two or more elements are in parallel if they are connected to the same two nodes, giving them the exact same voltage across their terminals.

• Worked Example: Identifying Topology Features

  • Problem: Determine the number of branches and nodes in a circuit containing a $10\,\text{V}$ source, a $5\,\Omega$ resistor, a $6\,\Omega$ resistor, and a $2\,\text{A}$ source. Identify series and parallel combinations.
  • Branches ($b$): $4$ total branches ($10\,\text{V}$, $5\,\Omega$, $6\,\Omega$, $2\,\text{A}$).
  • Nodes ($n$): $3$ total nodes.
  • Connection types:
  • The $5\,\Omega$ resistor is in series with the $10\,\text{V}$ voltage source because the same current flows through both.
  • The $6\,\Omega$ resistor is in parallel with the $2\,\text{A}$ current source because both elements are connected directly across the same pair of nodes.

Kirchhoff's Laws

• Kirchhoff's Current Law (KCL):

  • Principle: Based on the law of conservation of charge.
  • Statement: The algebraic sum of currents entering a node (or a closed boundary) is zero.     $\sum_{k=1}^N i_k = 0$
  • Alternative Statement: Sum of currents entering a node equals the sum of currents leaving the node.     $i_1 + i_3 + i_4 = i_2 + i_5$

• Kirchhoff's Voltage Law (KVL):

  • Principle: Based on the principle of conservation of energy.
  • Statement: The algebraic sum of all voltages around any closed path (or loop) is zero.     $\sum_{m=1}^M v_m = 0$
  • Alternative Statement: Sum of voltage drops equals the sum of voltage rises.     $v_2 + v_3 + v_5 = v_1 + v_4$

• Worked Example 1: Single Loop Analysis

  • Problem: Find voltages $v_1$ and $v_2$ in a circuit containing a $20\,\text{V}$ voltage source, a $2\,\Omega$ resistor ($v_1$), and a $3\,\Omega$ resistor ($v_2$ with reverse polarity).
  • Ohm's Law expressions:     $v_1 = 2 \times i$$v_2 = -3 \times i$
  • KVL around the loop:     $-20 + v_1 - v_2 = 0$
  • Substitute Ohm's law expressions into KVL:     $-20 + 2 \times i - (-3 \times i) = 0$$5 \times i = 20 \implies i = 4\,\text{A}$
  • Voltages calculated:     $v_1 = 2 \times 4 = 8\,\text{V}$$v_2 = -3 \times 4 = -12\,\text{V}$

• Worked Example 2: Multi-Loop Nodal Analysis

  • Problem: Calculate node voltage $v_1$ and branch currents ($i_1, i_2, i_3$) for a network with a $5\,\text{V}$ source, $2\,\Omega$ resistor, $8\,\Omega$ resistor, $4\,\Omega$ resistor, and an inverted $3\,\text{V}$ source.
  • Nodal Current Equations:     $i_1 = \frac{v_1 - 5}{2}$$i_2 = i_{8\Omega} = \frac{v_1 - 0}{8} = \frac{v_1}{8}$$i_3 = \frac{v_1 - (-3)}{4} = \frac{v_1 + 3}{4}$
  • KCL Equation at central node $v_1$:     $\frac{v_1 - 5}{2} + \frac{v_1}{8} + \frac{v_1 + 3}{4} = 0$
  • Multiplying through by $8$:     $(4 \times v_1 - 20) + (v_1) + (2 \times v_1 + 6) = 0$$7 \times v_1 = 14 \implies v_1 = 2\,\text{V}$
  • Branch Currents Calculated:     $i_1 = \frac{2 - 5}{2} = -1.5\,\text{A}$$i_2 = i_{8\Omega} = \frac{2}{8} = 0.25\,\text{A}$$i_3 = \frac{2 + 3}{4} = 1.25\,\text{A}$
  • KCL Verification:     $i_1 + i_2 + i_3 = -1.5 + 0.25 + 1.25 = 0\,\text{A}$
  • KVL Loop Checks:
  • Loop 1:       $-5 + v_{2\Omega} + v_{8\Omega} = -5 + [-(-1.5) \times 2] + (0.25 \times 8) = -5 + 3 + 2 = 0\,\text{V}$
  • Loop 2:       $-v_{8\Omega} + v_{4\Omega} - 3 = -(0.25 \times 8) + (1.25 \times 4) - 3 = -2 + 5 - 3 = 0\,\text{V}$

Series Circuits and Voltage Division

• Series Resistors Equivalent Resistance:   

  • Applying Ohm's law to two series resistors:     $v_1 = i \times R_1$$v_2 = i \times R_2$
  • KVL around the loop:     $-v + v_1 + v_2 = 0 \implies v = i \times (R_1 + R_2)$
  • Equivalent Resistance ($R_{\text{eq}}$):     $R_{\text{eq}} = R_1 + R_2$
  • General formula for $N$ resistors connected in series:     $R_{\text{eq}} = R_1 + R_2 + \dots + R_N = \sum_{n=1}^N R_n$

• Voltage Division Principle:

  • Voltage across $R_1$:     $v_1 = \frac{R_1}{R_1 + R_2} \times v$
  • Voltage across $R_2$:     $v_2 = \frac{R_2}{R_1 + R_2} \times v$
  • General voltage divider rule for the $n\text{-th}$ resistor in a series string of $N$ resistors:     $v_n = \frac{R_n}{R_1 + R_2 + \dots + R_N} \times v$

Parallel Circuits and Current Division

• Parallel Resistors Equivalent Resistance:   

  • For two resistors in parallel, voltage across each is identical ($v$):     $i_1 = \frac{v}{R_1}$$i_2 = \frac{v}{R_2}$
  • Applying KCL at the top node:     $i = i_1 + i_2 = v \times \left(\frac{1}{R_1} + \frac{1}{R_2}\right) = \frac{v}{R_{\text{eq}}}$
  • Reciprocal resistance sum:     $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{R_1 + R_2}{R_1 \times R_2}$
  • Equivalent resistance for two parallel resistors:     $R_{\text{eq}} = \frac{R_1 \times R_2}{R_1 + R_2}$(The equivalent resistance of two parallel resistors equals their product divided by their sum).

• Current Division Principle:

  • Branch current $i_1$:     $i_1 = \frac{R_2}{R_1 + R_2} \times i$
  • Branch current $i_2$:     $i_2 = \frac{R_1}{R_1 + R_2} \times i$
  • Key takeaway: Total current $i$ divides among parallel resistors in inverse proportion to their resistance values; larger current flows through the smaller resistance.

• Extreme Parallel Configurations:   

  • Shorted Parallel Branch ($R_2 = 0$): Entire current flows through the short circuit branch ($i_2 = i$), leaving zero current in the parallel resistor ($i_1 = 0$).
  • Open Parallel Branch ($R_2 = \infty$): Zero current flows through the open branch ($i_2 = 0$), forcing all current through the parallel resistor ($i_1 = i$).

• Worked Example: Complex Network Equivalent Resistance

  • Problem: Find $R_{\text{eq}}$ for the ladder network shown.
  • Step 1: Combine parallel resistors $6\,\Omega$ and $3\,\Omega$:     $6\,\Omega \parallel 3\,\Omega = \frac{6 \times 3}{6 + 3} = 2\,\Omega$
  • Step 2: Combine series resistors $1\,\Omega$ and $5\,\Omega$ in the rightmost branch:     $1\,\Omega + 5\,\Omega = 6\,\Omega$
  • Step 3: Combine series resistors in middle branch ($2\,\Omega$ resistor in series with the $2\,\Omega$ parallel combination from Step 1):     $2\,\Omega + 2\,\Omega = 4\,\Omega$
  • Step 4: Combine parallel combination of the middle branch ($4\,\Omega$) and right branch ($6\,\Omega$):     $4\,\Omega \parallel 6\,\Omega = \frac{4 \times 6}{4 + 6} = 2.4\,\Omega$
  • Step 5: Sum final series components ($4\,\Omega$, $2.4\,\Omega$, and $8\,\Omega$):     $R_{\text{eq}} = 4\,\Omega + 2.4\,\Omega + 8\,\Omega = 14.4\,\Omega$

Electromagnetic Induction

• Definition: The phenomenon of producing an electromotive force (EMF) or electric current in a conductor when the magnetic flux linked with the conductor changes over time. Occurs without any direct electrical contact between the magnetic field source and the conductor. Discovered by Michael Faraday in 1831.

• Physical Mechanism: When a conductor is exposed to a changing magnetic field, free electrons in the material experience a magnetic force causing them to move. This collective movement generates an induced EMF and, in a closed circuit, an induced current.

• Methods to Alter Magnetic Flux:

  1. Moving a conductor physically through a stationary magnetic field.
  2. Moving a magnet towards or away from a stationary conductor.
  3. Changing the intensity/strength of the magnetic field over time.
  4. Altering the effective area of a coil exposed to a magnetic field.
  5. Rotating a coil inside a uniform magnetic field.

• Magnetic Flux ($\Phi$):

  • Definition: The total number of magnetic field lines passing through a specified surface area.
  • Mathematical Expression:     $\Phi = B \times A \times \cos(\theta)$
  • Variables:
  • $\Phi$: Magnetic flux in Webers ($\text{Wb}$)
  • $B$: Magnetic field flux density/strength in Tesla ($\text{T}$)
  • $A$: Area of the coil in square meters ($\text{m}^2$)
  • $\theta$: Angle between the magnetic field direction and the normal vector perpendicular to the coil surface.

• Faraday's Laws of Electromagnetic Induction:

  • First Law: Whenever the magnetic flux linked with a conductor or coil changes, an EMF is induced in that conductor.
  • Second Law: The magnitude of the induced EMF is directly proportional to the time rate of change of magnetic flux linked with the circuit.
  • Governing Equation:     $E = -N \times \frac{d\Phi}{dt}$
  • Where $E$ is induced EMF in volts ($\text{V}$), $N$ is the number of turns in the coil, and $\frac{d\Phi}{dt}$ is the rate of flux change.

• Lenz's Law:

  • Accounted for by the negative sign in Faraday's equation.
  • Statement: The direction of the induced EMF/current is always such that its magnetic effect opposes the underlying change in flux that produces it.
  • Direct manifestation of the law of conservation of energy.
  • Practical Example: When the North pole of a magnet approaches a coil, the induced current flows in a direction that creates an opposing North pole at that end of the coil to resist motion. Conversely, when pulling the North pole away, the current reverses to create a South pole to attract it.

• Induction Types:

  • Self-Induction: A phenomenon where a time-varying current in a coil induces an opposing EMF within the same coil (e.g., inductors, chokes).
  • Mutual Induction: A phenomenon where a time-varying current in a primary coil induces an EMF in a nearby secondary coil (e.g., transformers).

• Factors Affecting Induced EMF Magnitude:

  1. Number of wire turns in the coil ($N$)
  2. Magnetic field strength ($B$)
  3. Relative velocity between conductor and field ($v$)
  4. Surface area of the coil ($A$)
  5. Spatial angle between magnetic field and coil plane ($\theta$)

• Fleming's Right-Hand Rule:   

  • Purpose: Used to determine the direction of induced current in moving conductors, primarily applied in electric generators.
  • Orientation: Extend the thumb, forefinger, and middle finger of the right hand mutually perpendicular (at $90^\circ$ angles) to one another:
  • Thumb: Points in the direction of Motion or Thrust of the conductor relative to the magnetic field.
  • Forefinger: Points in the direction of the Magnetic Field.
  • Middle Finger: Points in the direction of the Induced Current.

Statically vs. Dynamically Induced EMF

• Statically Induced EMF:

  • Definition: EMF induced in a conductor that remains physically stationary while the surrounding magnetic field varies with time.
  • Mechanism: The conductor/coil is stationary; the magnetic field strength fluctuates over time.
  • Operating Principle: Faraday's law achieved via a time-varying magnetic field (e.g., alternating current).
  • Formula:     $e = -N \times \frac{d\Phi}{dt}$
  • Primary Application: Transformers.

• Dynamically Induced EMF:

  • Definition: EMF induced in a conductor that physically moves through a uniform, stationary magnetic field, cutting across magnetic lines of force.
  • Mechanism: The magnetic field is stationary; the conductor moves physically to cut flux lines.
  • Operating Principle: Faraday's law achieved through mechanical motion creating relative velocity between field and conductor.
  • Formula:     $e = B \times l \times v \times \sin(\theta)$
  • Where $B$ is magnetic flux density, $l$ is active conductor length, $v$ is linear motion velocity, and $\theta$ is the angle of motion relative to the magnetic field.
  • Primary Application: Electric generators (both AC and DC) and electric motors.

• Summary Comparison Table: